Trigonometry for US Curriculum

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Trigonometry is usually taught with either Algebra 2, Geometry, or Pre-Calculus. However, some schools in the US teach Trigonometry as a separate course. Best Online Tutors for Trigonometry from India are available on Noble Learners. Book your free demo now!

Trigonometry Syllabus

The chapters given below is the part of syllabus which we will discuss in course. In case of any variation in the syllabus our expert tutors from India will teach as per the need of the course from one student to another.

Chapter 1: Trigonometric Ratios

Chapter 2: Trigonometric Angles

Chapter 3: Trigonometric Simplification.

Chapter 4: Trigonometric Identities.

Chapter 5: Conversion of Degree into Radians.

Chapter 6: Co-terminal and Reference Angles.

Chapter 7: Nature of Trigonometric functions in Quadrants.

Chapter 8: Sine and Cosine Rule

Chapter 9: More about Inverse Trigonometric functions.

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Online Trigonometry Tutor from India for USA

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Chapter 1: Trigonometric Ratios

  1. Question: Given that \( \sin(x) = \frac{3}{5} \), find \( \cos(x) \).
    Answer: \( \cos(x) = \frac{4}{5} \).
  2. Question: If \( \tan(y) = \frac{7}{24} \), find \( \csc(y) \).
    Answer: \( \csc(y) = \frac{24}{7} \).
  3. Question: Find the value of \( \sin(\theta) \) if \( \cos(\theta) = -\frac{3}{5} \).
    Answer: \( \sin(\theta) = -\frac{4}{5} \).
  4. Question: Given that \( \csc(z) = \frac{5}{3} \), find \( \sec(z) \).
    Answer: \( \sec(z) = \frac{3}{4} \).
  5. Question: If \( \cot(\alpha) = -\frac{5}{12} \), find \( \tan(\alpha) \).
    Answer: \( \tan(\alpha) = -\frac{12}{5} \).

Chapter 2: Trigonometric Angles

  1. Question: Find the exact value of \( \sin(45^\circ) \).
    Answer: \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \).
  2. Question: Calculate \( \cos(60^\circ) \).
    Answer: \( \cos(60^\circ) = \frac{1}{2} \).
  3. Question: Determine \( \tan(30^\circ) \).
    Answer: \( \tan(30^\circ) = \frac{\sqrt{3}}{3} \).
  4. Question: Find \( \sec(120^\circ) \).
    Answer: \( \sec(120^\circ) = -2 \).
  5. Question: Calculate \( \cot(45^\circ) \).
    Answer: \( \cot(45^\circ) = 1 \).

Chapter 3: Trigonometric Simplification

  1. Question: Simplify \( \sin^2(x) + \cos^2(x) \).
    Answer: \( \sin^2(x) + \cos^2(x) = 1 \).
  2. Question: Simplify \( \frac{1 + \tan^2(x)}{\sec^2(x)} \).
    Answer: \( \frac{1 + \tan^2(x)}{\sec^2(x)} = 1 \).
  3. Question: Simplify \( \frac{\csc^2(x)}{\cot^2(x)} \).
    Answer: \( \frac{\csc^2(x)}{\cot^2(x)} = \sin^2(x) \).
  4. Question: Simplify \( \frac{1 - \sin^2(x)}{1 + \cos^2(x)} \).
    Answer: \( \frac{1 - \sin^2(x)}{1 + \cos^2(x)} = \cos^2(x) \).
  5. Question: Simplify \( \frac{\tan^2(x)}{1 - \cot^2(x)} \).
    Answer: \( \frac{\tan^2(x)}{1 - \cot^2(x)} = \sin^2(x) \).

Chapter 4: Trigonometric Identities

  1. Question: Prove the identity: \( \sin^2(x) + \cos^2(x) = 1 \).
    Answer: The identity is a fundamental trigonometric identity known as the Pythagorean identity.
  2. Question: Simplify \( \sin(x) \csc(x) \).
    Answer: \( \sin(x) \csc(x) = 1 \).
  3. Question: Prove the identity: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
    Answer: This identity follows directly from the definitions of sine and cosine.
  4. Question: Simplify \( \cos^2(x) - \sin^2(x) \).
    Answer: \( \cos^2(x) - \sin^2(x) = \cos(2x) \).
  5. Question: Prove the identity: \( \tan(x) \cdot \cot(x) = 1 \).
    Answer: Multiply \( \tan(x) \) and \( \cot(x) \) and simplify to get \( 1 \).

Chapter 5: Conversion of Degree into Radians

  1. Question: Convert \( 45^\circ \) to radians.
    Answer: \( 45^\circ = \frac{\pi}{4} \) radians.
  2. Question: Convert \( 120^\circ \) to radians.
    Answer: \( 120^\circ = \frac{2\pi}{3} \) radians.
  3. Question: Convert \( 300^\circ \) to radians.
    Answer: \( 300^\circ = \frac{5\pi}{3} \) radians.
  4. Question: Convert \( 210^\circ \) to radians.
    Answer: \( 210^\circ = \frac{7\pi}{6} \) radians.
  5. Question: Convert \( 315^\circ \) to radians.
    Answer: \( 315^\circ = \frac{7\pi}{4} \) radians.

Chapter 6: Co-terminal and Reference Angles

  1. Question: Find the reference angle for \( -150^\circ \).
    Answer: The reference angle is \( 30^\circ \).
  2. Question: Determine the co-terminal angle for \( \frac{5\pi}{6} \).
    Answer: \( \frac{5\pi}{6} \) and \( \frac{17\pi}{6} \) are co-terminal angles.
  3. Question: Find the reference angle for \( \frac{11\pi}{4} \).
    Answer: The reference angle is \( \frac{\pi}{4} \).
  4. Question: Determine the co-terminal angle for \( -\frac{3\pi}{2} \).
    Answer: \( -\frac{3\pi}{2} \) and \( \frac{\pi}{2} \) are co-terminal angles.
  5. Question: Find the reference angle for \( -\frac{7\pi}{3} \).
    Answer: The reference angle is \( \frac{\pi}{3} \).

Chapter 7: Nature of Trigonometric Functions in Quadrants

  1. Question: In which quadrant(s) is the sine function positive?
    Answer: The sine function is positive in quadrants I and II.
  2. Question: In which quadrant(s) is the tangent function negative?
    Answer: The tangent function is negative in quadrants II and IV.
  3. Question: In which quadrant(s) is the cosine function negative?
    Answer: The cosine function is negative in quadrants II and III.
  4. Question: Determine the quadrant(s) where the cosecant function is positive.
    Answer: The cosecant function is positive in quadrants I and II.
  5. Question: Identify the quadrant(s) where the secant function is negative.
    Answer: The secant function is negative in quadrants II and III.

Chapter 8: Sine and Cosine Rule

  1. Question: Apply the sine rule to solve for side \( c \) in triangle \( ABC \), given \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \).
    Answer: Use the formula \( \frac{c}{\sin(C)} = \frac{b}{\sin(B)} \) to find \( c \).
  2. Question: Use the cosine rule to find the length of side \( a \) in triangle \( ABC \), given \( a^2 = b^2 + c^2 - 2bc \cos(A) \).
    Answer: Substitute the given values to find \( a \).
  3. Question: Apply the sine rule to find the measure of angle \( B \) in triangle \( ABC \), given \( \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \).
    Answer: Use the formula \( \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \) to find \( \sin(B) \), then use inverse sine to find \( B \).
  4. Question: Use the cosine rule to solve for angle \( C \) in triangle \( ABC \), given \( c^2 = a^2 + b^2 - 2ab \cos(C) \).
    Answer: Substitute the given values to find \( C \).
  5. Question: Apply the sine rule to solve for side \( b \) in triangle \( ABC \), given \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \).
    Answer: Use the formula \( \frac{b}{\sin(B)} = \frac{a}{\sin(A)} \) to find \( b \).

Chapter 9: More about Inverse Trigonometric Functions

  1. Question: Find the exact value of \( \arcsin\left(\frac{\sqrt{3}}{2}\right) \).
    Answer: \( \arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \).
  2. Question: Determine \( \arccos(0) \).
    Answer: \( \arccos(0) = \frac{\pi}{2} \).
  3. Question: Find \( \arctan\left(-\sqrt{3}\right) \).
    Answer: \( \arctan\left(-\sqrt{3}\right) = -\frac{\pi}{3} \).
  4. Question: Find \( \arccos\left(-\frac{1}{2}\right) \).
    Answer: \( \arccos\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \).